The first $3$ terms in the expansion of $(1 + ax)^n$ $(n \ne 0)$ are $1, 6x$ and $16x^2$. Then the value of $a$ and $n$ are respectively

$(\sqrt{2} + 1)^6 - (\sqrt{2} - 1)^6 = $

The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3) \ldots (1+x^{100})$ is

Assertion $(A)$: If $a_1, a_2, \ldots, a_n$ are the $n$ distinct roots of the equation $x^n-2=0$,then $1+\left(1-a_1\right)\left(1-a_2\right) \ldots\left(1-a_n\right)=0$.
Reason $(R)$: If $\alpha_1, \alpha_2, \ldots, \alpha_n$ are the roots of $f(x) \equiv p_0 x^n+p_1 x^{n-1}+\ldots+p_n=0$,then the roots of $f(g(x))=0$ are $g^{-1}(\alpha_i)$ for $i=1, 2, \ldots, n$.
The correct option among the following is:

The sum of the coefficients in the expansion of $(1 + x - 3x^2)^{2163}$ will be

Find an approximation of $(0.99)^{5}$ using the first three terms of its binomial expansion.